PreprintCC BY-SA 4.0arXiv
Conjugate radius and parabolicity along the Ricci flow
Abstract
We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if (Mn,g(t))t∈[0,T] is a complete Ricci flow with nonnegative scalar curvature at t=0, the initial metric is parabolic and not Ricci-flat, and the Einstein tensor Ric−1/2 Rg is bounded above, then parabolicity is preserved along the flow and the conjugate radius of g(t) is finite for every t∈(0,T]. As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that liminfr→∞Vol(B(p,r))/r2=0, has finite conjugate radius.
Related
BibTeX
@misc{mondal2026conjugate,
title = {{Conjugate radius and parabolicity along the Ricci flow}},
author = {Chandan Kumar Mondal},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03823},
eprint = {2611.03823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03823},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Conjugate radius and parabolicity along the Ricci flow
AU - Chandan Kumar Mondal
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03823
UR - https://arxiv.org/abs/2611.03823
AB - We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if (M^n,g(t))_t∈[0,T] is a complete Ricci flow with nonnegative scalar curvature at t=0, the initial metric is parabolic and not Ricci-flat, and the Einstein tensor Ric−1/2 Rg is bounded above, then parabolicity is preserved along the flow and the conjugate radius of g(t) is finite for every t∈(0,T]. As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that liminf_r→∞Vol(B(p,r))/r^2=0, has finite conjugate radius.
ER -